# dependence on initial conditions of solutions of ordinary differential equations

Let $E\subset W$ where $W$ is a normed vector space, $f\in C^{1}(E)$ is a continuous differentiable map $f:E\to W$. Furthermore consider the ordinary differential equation

 $\dot{x}=f(x)$

with the initial condition

$x(0)=x_{0}$.

Let $x(t)$ be the solution of the above initial value problem defined as

 $x:I\to E$

where $I=[-a,a]$. Then there exist $\delta>0$ such that for all $y_{0}\in N_{\delta}(x_{0})$($y_{0}$ in the $\delta$ neighborhood of $x_{0}$) has a unique solution $y(t)$ to the initial value problem above except for the initial value changed to $x(0)=y_{0}$. In addition $y(t)$ is twice continouously differentialble function of $t$ over the interval $I$.

Title dependence on initial conditions of solutions of ordinary differential equations DependenceOnInitialConditionsOfSolutionsOfOrdinaryDifferentialEquations 2013-03-22 13:37:19 2013-03-22 13:37:19 Daume (40) Daume (40) 7 Daume (40) Theorem msc 35-00 msc 34-00